Infinity Essays - Cardinal Numbers, Infinity, Mathematical Objects

Infinity Essays - Cardinal Numbers, Infinity, Mathematical Objects Infinity Most everyone is familiar with the infinity symbol, the one that looks like the number eight tipped over on its side. Infinity sometimes crops up in everyday speech as a superlative form of the word many. But how many is infinitely many? How big is infinity? Does infinity really exist? You can't count to infinity. Yet we are comfortable with the idea that there are infinitely many numbers to count with; no matter how big a number you might come up with, someone else can come up with a bigger one; that number plus one, plus two, times two, and many others. There simply is no biggest number. You can prove this with a simple proof by contradiction. Proof: Assume there is a largest number, n. Consider n+1. n+1*n. Therefore the statement is false and its contradiction, there is no largest integer, is true. This theorem is valid based on the Validity of Proof by Contradiction. In 1895, a German mathematician by the name of Georg Cantor introduced a way to describe infinity using number sets. The number of elements in a set is called its cardinality. For example, the cardinality of the set 3, 8, 12, 4} is 4. This set is finite because it is possible to count all of the elements in it. Normally, cardinality has been detected by counting the number of elements in the set, but Cantor took this a step farther. Because it is impossible to count the number of elements in an infinite set, Cantor said that an infinite set has No elements; By this definition of No, No+1=No. He said that a set like this is countable infinite, which means that you can put it into a 1-1 correspondence. A 1-1 correspondence can be seen in sets that have the same cardinality. For example, 1, 3, 5, 7, 9}has a 1-1 correspondence with 2, 4, 6, 8, 10}. Sets such as these are countable finite, which means that it is possible to count the elements in the set. Cantor took the idea of 1-1 correspondence a step farther, though. He said that there is a 1-1 correspondence between the set of positive integers and the set of positive even integers. E.g. 1, 2, 3, 4, 5, 6, ...n ...} has a 1-1 correspondence with 2, 4, 6, 8, 10, 12, ...2n ...}. This concept seems a little off at first, but if you think about it, it makes sense. You can add 1 to any integer to obtain the next one, and you can also add 2 to any even integer to obtain the next even integer, thus they will go on infinitely with a 1-1 correspondence. Certain infinite sets are not 1-1, though. Canter determined that the set of real numbers is uncountable, and they therefore can not be put into a 1-1 correspondence with the set of positive integers. To prove this, you use indirect reasoning. Proof: Suppose there were a set of real numbers that looks like as follows 1st 4.674433548... 2nd 5.000000000... 3rd 723.655884543... 4th 3.547815886... 5th 17.08376433... 6th 0.00000023... and so on, were each decimal is thought of as an infinite decimal. Show that there is a real number r that is not on the list. Let r be any number whose 1st decimal place is different from the first decimal place in the first number, whose 2nd decimal place is different from the 2nd decimal place in the 2nd number, and so on. One such number is r=0.5214211... Since r is a real number that differs from every number on the list, the list does not contain all real numbers. Since this argument can be used with any list of real numbers, no list can include all of the reals. Therefore, the set of all real numbers is infinite, but this is a different infinity from No. The letter c is used to represent the cardinality of the reals. C is larger than No. Infinity is a very controversial topic in mathematics. Several arguments were made by a man named Zeno, a Greek mathematician who lived about 2300 years ago. Much of Cantors work tries to disprove his theories. Zeno said, There is no motion because that which moved must arrive at the middle of its course before it arrives at the end.

One Art Essays

One Art Essays One Art Paper One Art Paper Essay Topic: Poetry The art of losing isnt hard to master; so many things seem filled with the intent to be lost that their loss is no disaster, Lose something every day. Accept the fluster of lost door keys, the hour badly spent. The art of losing isnt hard to master. Then practice losing farther, losing faster: places, and names, and where it was you meant to travel. None of these will bring disaster. I lost my mothers watch. And look! my last, or next-to-last, of three beloved houses went. The art of losing isnt hard to master. I lost two cities, lovely ones. And, vaster, some realms I owned, two rivers, a continent. I miss them, but it wasnt a disaster. Even losing you (the joking voice, a gesture I love) I shant have lied. Its evident the art of losings not too hard to master though it may look like (Write it!) a disaster. Why I chose it She explains a major problem well in this poem. We have so many losses in life and we have to learn how to forget some of them otherwise life will overwhelm us. Only if we do this, we will learn how to deal with the big losses in life and the pain that comes with them. Elizabeth Bishop is trying to teach this important lesson to the reader and she does this by creating a poem with an interesting structure. She increases the amount that is lost after each stanza so the reader can understand how forgetting small losses can help them get over bigger losses. She uses a rhyme scheme of ABA in each of her 3 line stanzas and each middle line of the stanzas rhyme with each other middle line of the other stanzas e.g went, continent, intent. With this, the poem flows nicely which helps the reader understand it. Also I was curious on the poem because of the title. After reading it I questioned how loss can be named an art. Bishop explains this clearly in her poem. Commentary Elizabeth Bishops poem, One Art, is a poem overflowing with irony which uses the structure and expressions to evoke the emotion of loss which Bishop is trying to convey to the reader. She seems to have lost many things in her life and has written this poem for the sole purpose to reassure the reader that if they find control within themselves, they are able to accept the many losses that are in their lives. Loss is something that cannot be mastered, emotion is always attached to loss. But Bishop is trying to tell the reader that with each loss, you endure it more and will ultimately learn to control the feeling, with less pain. The speaker in this poem seems to be Bishop dealing with her inner emotions of loss, trying to master loss so she does not have to deal with the painful emotions afterwards. This is ironic at the very concept of mastering loss, as loss is not a feeling that can be mastered. The structure of the poem is a villanelle as there are five stanzas of three lines followed by a stanza of four lines. Bishop seems to have utilised this structure to increase the readers attention and respect for loss as each stanza progresses. The tone of this poem is recognized by the reader from reading the first line. The poem deals with loss which is associated with death. A poem dealing with this theme will most certainly have a negative tone, and One Art is no exception. The tone changes with each stanza, gaining in sadness as the poem continues. This is because the value of what is lost increases within each stanza. This tone helps the reader emphasise on the value of loss. Bishop uses repetition often and refrains the line The art of losing isnt hard to master which is done to stress on the importance of the title. Also the readers sense of the subject is gained by each repetition as the theme of the poem is emphasised after each stanza. Each refrain builds up stronger amounts of loss. The second stanza is based on the loss of door keys or wasting time. The poet personifies these lost objects, saying that they seem filled with the intent to be lost. This is showing that these objects want to be lost and by giving inanimate objects feelings, it conveys that these objects are not of great importance. These are basically simple losses that one can learn to forget, but as the poem progresses, the amount that is lost builds up, including personal things with much more importance. This point can be proven in the third stanza, where she says Then practice losing farther, losing faster: This is the point in the poem, where she begins to name things that can be lost with much greater importance. Following the third stanza, the next one involves things of personal importance to a person. my mothers watchthree beloved houses. These things are more personal as a home is one which symbolises love and protection, which is a contrast to the insignificant loss of keys. In the fifth stanza, she describes the loss of things with huge importance, two rivers, a continent. Bishop continues the patter of possessions increasing in importance, by describing things which affect thousands of people. She seems to have reached things of the highest importance but finally in the last stanza, it all leads up to the most painful loss, the loss of a lover. After the fifth stanza, she uses a dash to start the last stanza. This is done to show that she is finished describing the loss of material objects, and is moving on to the loss of something with the most importance. Its evident the art of losings not too hard to master though it may look like (Write it!) a disaster. This entire stanza flows together due to the quicker pace and full use of enjambment throughout the entire stanza. The reader was use to the punctuation increasing and pace slowing down each stanza from the first to the fifth, so this increased the impact of pace on the last stanza, making it more powerful. Parenthesis was effectively used to show how hard and painful it was for the writer to write down her emotions into words, (Write it!). Bishop is trying to teach the reader to endure the many small losses, in order to feel less pain in the loss of something with much importance. By breaking away from describing the loss of an inanimate material objects towards a living person, she amplifies the feeling of loss which conveys the message of the poem clearly to the reader. With each stanza, we further understand the emotions which Bishop is trying to convey to us. She uses the title to imply how dealing with a loss is an art in its own accord. An art is something of beauty that requires massive amounts of effort to accomplish. The use of punctuation is another distinct feature of the structure of this poem, with each stanza, the amount of punctuation increases up until the fifth stanza. Bishop increases the value of what was being lost, along with this, the punctuation increases as well. This was done to slow down the pace of the poem so that the reader understands the importance of each possession. When describing each thing that was being lost in the fourth, two cities, lovely ones.two rivers, a continent. Along with the commas, caesura was used as well in the first line to slow down the pace. By forcing the reader to stop in the middle of the line, it takes them by surprise as it was sudden, not done in the poem before. This stanza contrasts with the quicker pace of the first stanza as there was only one semi-colon and comma used. This is clear evidence of the punctuation used by Bishop which greatly emphasises what Bishop was trying to achieve with this poem. This poem was written to show the readers the comparison of loss between different things. This poem may seem simple, but is very deep mainly through the way the poet utilised the structure and the richness in irony. This can be seen by how she refrains the use of loss isnt hard to master. This changes in the last line to loss isnt too hard to master This might show how it is getting harder for the poet to overcome the loss of a loved one. Bishop proves that the loss of trivial things and mastering this loss, will not be able to prepare you for the loss of something beloved by you, which further increases the ironic nature of the poem. In the end she could be all alone, hence the title One Art. She believed that if she could detach herself from pain and forget these feelings, she can begin a new life and grow, but realises that it is too hard for her.

Bentley University Acceptance Rate and SAT Score Facts

Bentley University Acceptance Rate and SAT Score Facts Bentley requires test scores as part of its admissions process. Students can submit either SAT or ACT scores, and neither is preferred over the other. Bentley is a selective school, and only admits 42 percent of applicants each year. To apply, students must fill out a Common Application, complete with the writing/personal statement section. In addition, students must submit test scores, an application fee, and high school transcripts. Bentleys admissions are holistic, meaning that they look at more than just grades and test scores. Students who participate in extracurricular activities and have work or volunteer experience are encouraged to list their activities and experience on their applications, to help set them apart. Will You Get In? Calculate Your Chances of Getting In  with Cappexs free tool. Admissions Data (2016) Bentley University Acceptance Rate: 46%GPA, SAT and ACT graph for Bentley Test Scores:  25th / 75th Percentile SAT Critical Reading: 510 / 620SAT Math: 550  / 670SAT Writing: - / -What these SAT numbers meanACT Composite: 25  / 30What these ACT numbers mean Bentley University Description Located on a 163-acre campus in Waltham, Massachusetts, Bentley University is not a typical New England college. The great majority of Bentley students major in some area of business, but the school is nevertheless a comprehensive university where the liberal arts and sciences play a central role in the curriculum. Ethics, social responsibility, and global culture are all important components of a Bentley business education. Bentley has a 12 to 1  student/faculty ratio  and an average class size of 24. The university frequently ranks among the top 50 business schools in the country. On the athletic front, the Bentley University Falcons compete in the NCAA Division II, within the Northeast-10 Conference. Popular sports include track and field, football, basketball, and soccer. Bentley University Mission Statement To educate creative, ethical, and socially responsible organizational leaders by creating and disseminating impactful knowledge within and across business and the arts and sciences. Enrollment (2016) Total Enrollment: 5,506  (4,222 undergraduates)Gender Breakdown: 59% Male / 41% Female98% Full-time Costs (2016 - 17) Tuition and Fees: $45,760Books: $1,260  (why so much?)Room and Board: $15,130Other Expenses: $1,200Total Cost: $61,005 Bentley University Financial Aid (2015  - 16) Percentage of New Students Receiving Aid: 76%Percentage of New Students Receiving Types of AidGrants: 68%Loans: 49%Average Amount of AidGrants: $25,343Loans: $8,540 Academic Programs: Most Popular Majors:  Accounting, Business Administration, Finance, Marketing, Computer Science, Mathematics, Global Studies, Business/Corporate CommunicationsWhat major is right for you?  Sign up to take the free My Careers and Majors Quiz at Cappex. Graduation and Retention Rates: First Year Student Retention (full-time students): 94%Transfer-out Rate: 3%4-Year Graduation Rate: 83%6-Year Graduation Rate: 89% Intercollegiate Athletic Programs: Mens Sports:  Football, Lacrosse, Soccer, Swimming, and Diving Track, and Field, Basketball, Ice Hockey, BaseballWomens Sports:  Field Hockey, Basketball, Track and Field, Softball, Tennis, Volleyball, Cross Country Bentley and the Common Application Bentley University uses the  Common Application. These articles can help guide you: Common Application essay tips and samplesShort answer tips and samplesSupplemental essay tips and samples Data Source: National Center for Educational Statistics

Economic Geography Essay Example | Topics and Well Written Essays - 500 words

Economic Geography - Essay Example vious recessions in 1990-1991 and 2001, the 2007 economic recession has run deep into the economy becoming the longest financial crisis since the great depression in the 1930s (Kaiser 3). Many states reacted by formulating large stimulus mechanisms in an effort to stabilize the economy. Other mechanisms were bailout of leading financial institutions, which included loans, guarantees and equity. The National Bureau of Economic Research and economists have indicated that the financial recession was triggered in the year 2006 by the broad economic malaise. The sharp deterioration in house prices reduced purchases of homes subsequently damaging the real estate market. Price deterioration extended to increased mortgage closure that subsequently led to massive losses in billions of dollars by leading banks in the country. The ultimate condition was tightening credit by financial institutions. The Federal Reserve Bank began to cut the discount rate and funds rate to no avail. Major Banks such as Lehman Brothers started filing for bankruptcy. Other banks opted to undertake mergers and acquisition. JPM Chase acquired Bear Stearns as a strategy to survive in the crippling economy. As indicated by Cadieux & Conklin (9), the Federal Reserve cut the two rates below 2% by the fall of October 2008. In the same line central banks of leading world economies like China, Canada and United Kingdom as well as the ECB slashed their rates to rescue the global economy. These efforts had little impact as liquidity problem that proved to have spread widely and damaged operation and of financial institutions. The rate of unemployment increased by 1.2% while an increasing number of distributions of jobless claims in the industry proved the economy was under recession (Cadieux & Conklin 12). The manufacturing and construction industries experienced large labor losses that significantly shattered business cycle. Although the level of unemployment declined as time moved to the end of

Curriculum Development Related to Information Security Policies and Case Study

Curriculum Development Related to Information Security Policies and Procedures - Case Study Example For providing improved functionality for the organization, policies and procedures must be defined. They play a vital role for an organization’s smooth functioning. In order to implement policies and procedures, group discussions are required for constructing and implementing them in a real world scenario. The first requirement is to differentiate both of them. A security policy comprises in the form of a document or rules that specify the statement ‘What must be done’ in order to assure security measures in the system or the network. Whereas, procedures are associated with the rules and practices that are implemented in order to impose the rule. For instance, in a network security scenario, where there is a requirement for preventing the wireless network, anonymous access must be blocked. Likewise, the security policy document will define ‘What needs to be done’ to block anonymous access for a wireless network. Whereas, the procedures will define the practices and rules that needs to be followed in order to block the anonymous access ("Curriculum development related to information security policies and procedures ").After differentiating both the security policies and procedures, these two are associated with development and administration in an organization. The term security in terms of development and administration is more like a management issue rather than a technical issue in an organization. The justification is to utilize and classify employees of an organization efficiently. Moreover, from the management perspective, discussions take place for describing various vulnerabilities and threats along with the creation of policies and procedures that may contribute for the achievement of organization goals. After the discussions and alignment of policies and procedures to contribute for organization’s success, the development process is initiated at a high level, and afterwards implemented at lower levels within an o rganization. The conclusion reflects the development of policies and procedures, requirement of an approval from concerned personnel and then implementing them smoothly for the employees ("Curriculum development related to information security policies and procedures "). On the other hand, initiation of these security policies is easy and not expensive, but the implementation is the most difficult aspect. If the development and administration do not comply effectively, or fails to establish awareness between employees related to the policies and procedures, the disadvantages may affect inadequately for the organization. For instance, an attack from a social engineering website such as ‘Facebook’, ‘twitter’, or ‘MySpace’ may extract sensitive information from senior or trusted employees of an organization. If the policies and procedures were understood or implemented properly, employees will be well aware of not providing any credentials or they will verify authorization before providing information on the sites. Moreover, privacy and trust is a debatable topic that is also referred as identity theft. A good definition of identity theft is available in network dictionary that states as, â€Å"Identity theft is a crime in which an imposter obtains key pieces of personal information, such as Social Security or driver’s license numbers, in order to impersonate someone else. The information can be used to obtain credit, merchandise, and services in the name of the victim, or to provide the thief with false credentials†. Many organizations have suffered security breaches initiated from a social networking site. There is a requirement for creating awareness between employees by developing policies and procedures related to the attacks associated with social engineering sites along with coping up these attacks. There is a requirement of identifying critical information systems within the organization. All the users must be

Did the frontierhelp shape American individualism Assignment

Did the frontierhelp shape American individualism - Assignment Example It took centuries of efforts and sacrifices before the country achieved its illustrious and commanding status in the world. If not for these events, it will just be another nation that would want to make a name for itself. For new frontiers to be discovered and developed, vast efforts were undertaken by generations of people who have dreamt of reaching not just fame for people as individuals but for the nation as a whole (Hoover, 2005). As Turner (1893) stated on his essay The Significance of the Frontier in American History, the frontier is the borderline of barbarism and development. The 300 years in the history of the United States and its people serve as the frontier of the land. Though the actions and strategies that were taken during these years were truly primeval compared to the ideologies and designs which are being used in recent years, they are responsible for many important details and events in history. The most important contribution or event by these primeval acts is the liberation of the land from its colonizers. This single event transformed a land to a civilized nation. Three hundred years of struggle can be considered as baby steps to the present pace of the country in different aspects of the society and governance. If not for these baby steps, the nation would not be capable of taking leaps that have made their mark in world history. What used to be desert and unplowed lands are now either cultivated fields or modernized cities. If there would be one thing that the present Americans should learn from their forefathers is the fact that during the three-century frontier, they fought for the land as a whole and not on a per state basis. This should be considered if the present nation would like to sustain its stature and the ability to adapt to changes that are being consistently undertaken not just by the Americans

Frederick douglass Essay Example | Topics and Well Written Essays - 1250 words

Frederick douglass - Essay Example Though he lived to see the official end of slavery, the struggle for equality continued and still does to this day inspired by his and many others bravery and sacrifice. The illegitimate son of a white plantation owner father and slave mother, the early years of Frederick Douglass, originally Frederick Augustus Washington Bailey, life was unstable. He was given to his grandmother to raise at the age of 10 following his mothers untimely death in about 1828. Soon after, Douglass was afforded the opportunity to live in the masters home, presumably because the owner acknowledged that he was his biological son. When his plantation owner, Captain Anthony, died Douglass could no longer live in a white mans house so he was sent to live with Anthonys sister Lucretia Auld who then gave him to her Baltimore brother-in-law Hugh Auld. Barely a teenager, Douglass found himself in another new setting but this time surrounded by nothing but strangers. He turned this uneasy situation into a positive outcome by learning how to read and write from area children of white families though he had to keep it a secret from his Master Auld who forbade him from receiving an e ducation. Auld knew that if slaves learned to read they would become inspired by anti-slave literature which was the case for Douglass who attributed the Columbian Orator for inspiring his views on human rights. (Lewis, 2014). During his teen years Douglass was sold, transferred or hired-out to several other slave owners. While working for William Freedland, Douglass secretly taught class on the plantation which was faithfully attended by as many as 40 other slaves, a practice tolerated by Freedland but not by locals who, armed with various weapons, violently raided the make-shift classroom and ensured the clandestine practice ended. While working for Edward Covey, a slave owner with a brutal reputation, the young